You could save the returned results in a state. So press your button, call the API, get results, put the result in a state of an element, show what is saved in the state.
Do you mean the validation response to know if the postcode is valid e.g. yes/no?
This can be saved in a custom state on page no problem. Same could be set with the postcode as either text or a geographical location field type, depends what you’ll do with it.
Saving all the postcodes lookup wouldn’t be possible, turns out there are over 1.7m variations